Past exam of the lie Groups and Differentiable Manifolds course of the University of Oxford / 2008 / A / 3

With the convention applicable also to disconnected compact groups, the Weyl group is $W=N_G(T)/T$, where the normalizer consists of elements taking $T$ to itself by conjugation. Put $\mathfrak{t}=L(T)$ and $\Lambda =\ker (\exp :\mathfrak{t}\to T)$. This is a full lattice with $T\cong \mathfrak{t}/\Lambda$.

If $\psi$ is a torus automorphism, exponential compatibility gives $\psi (\exp X)=\exp (D{\psi }_{I}X)$. Therefore $D{\psi }_I$ sends $\Lambda$ into itself; applying the inverse proves it sends the lattice onto itself. In a lattice basis this derivative is an element of $G{L}_r(\mathbb{Z})$.

Conjugation by a connected subgroup $Q\subset N_G(T)$ induces a continuous family of these integer matrices. Since $G{L}_r(\mathbb{Z})$ is discrete, this family is constant and equals its value at the identity. Its action on $\mathfrak{t}$ is therefore trivial, and surjectivity of the torus exponential shows that $Q$ centralizes $T$.

Apply this to the identity component $N_G(T{)}^0$. It lies in $G^0$ and centralizes $T$. The allowed maximal-abelian property of $T$ in $G^0$ gives $N_G(T{)}^0\subseteq T$, while the connected group $T$ lies in that identity component. Thus $N_G(T{)}^0=T$. The normalizer is a closed compact Lie subgroup, so its component quotient is discrete and compact, hence finite. Consequently $W$ is finite and discrete. For disconnected $G$, its action on the lattice need not be faithful; the component argument is essential.

The distinction is recorded in Weyl group of a disconnected compact Lie group.

A finite discrete group has trivial identity component, so its only torus is $T=\{1\}$. Thus $N_G(T)=G$ and, under the definition $N_G(T)/T$,The alternative quotient by the centralizer would give a different answer for a disconnected group; it is not the convention used here.

A maximal torus is $T=S{O}_2$, which is normal of index two in $O_2$. ThereforeA reflection represents its nontrivial element and conjugates a rotation of angle $\theta$ to the rotation of angle $-\theta$.

Use the given torus of independent rotations in $n$ mutually orthogonal two-planes and a fixed final axis. A normalizer element permutes the two-plane rotation characters, and on each plane either preserves or reverses its orientation. Its induced action therefore permutes the $n$ angle coordinates and changes their signs independently.

Every such signed permutation is realized. Permutations of two-dimensional blocks have determinant one. A reflection in any chosen plane reverses that angle; reflecting the remaining one-dimensional axis as necessary restores total determinant one. The kernel consists precisely of independent plane rotations: commuting with a generic torus element forces preservation of its distinct rotation planes, and the determinant condition then fixes the final axis. Thus the kernel is $T$ andThis is the signed permutation group, the Weyl group of type $B_n$.